Ground states of the defocusing NLSE with a point interaction

Abstract

Suppose that either (i) $N=2$, $\alpha \in \mathbb{R}$ and $p>2$ or (ii) $N=3$, $\alpha <0$ and $2<p<3$. We prove that there exists an explicitly computable ${\mu }_0={\mu }_0(N,\alpha ,p)>0$ such that if $0<\mu <{\mu }_0$, then the following normalized semilinear elliptic problem with a point interaction admits ground states: \[ \begin{cases} - Δ_αu + ωu + u |u|^{p - 2} = 0 &\text{in} ~ \mathbb{R}^N; \\ \|u\|_{\mathscr{L}^2}^2 = μ, \end{cases} \] where $-{\Delta }_{\alpha }$ denotes the Laplacian of point interaction (centered at the origin) with inverse scattering length $-2(N-1)\pi \alpha$ and we want to solve for $\omega \in \mathbb{R}$, $u:{\mathbb{R}}^N\to \mathbb{R}$. We remark that this kind of solutions does not exist in the framework of the defocusing NLSE without a point interaction.

Irrecoverable failure in proposed proof. We apologize for the interested readers